Manifolds with group actions and elliptic operators
Author(s)
Bibliographic Information
Manifolds with group actions and elliptic operators
（Memoirs of the American Mathematical Society, v. 540）
American Mathematical Society, 1994
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Note
"November 1994, volume 112, number 540 (end of volume)"--T.p
Includes bibliographical references (p. 76-78)
Description and Table of Contents
Description
This work studies equivariant linear second order elliptic operators P on a connected noncompact manifold X with a given action of a group G. The action is assumed to be cocompact, meaning that GV=X for some compact subset V of X. The aim is to study the structure of the convex cone of all positive solutions of Pu=0. It turns out that the set of all normalized positive solutions which are also eigenfunctions of the given G -action can be realized as a real analytic submanifold *G[0 of an appropriate topological vector space *H. When G is finitely generated, *H has finite dimension, and in nontrivial cases *G[0 is the boundary of a strictly convex body in *H. When G is nilpotent, any positive solution u can be represented as an integral with respect to some uniquely defined positive Borel measure over *G[0. Lin and Pinchover also discuss related results for parabolic equations on X and for elliptic operators on noncompact manifolds with boundary.
Table of Contents
Introduction Some notions connected with group actions Some notions and results connected with elliptic operators Elliptic operators and group actions Positive multiplicative solutions Nilpotent groups: extreme points and multiplicative solutions Nonnegative solutions of parabolic equations Invariant operators on a manifold with boundary Examples and open problems Appendix: analyticity of $\Lambda (\xi, \scr L)$ References.
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